I would definitely not recommend Parenscript to a Lisp beginner. Parenscript has several features which will be extremely confusing for a new Lisp programmer. It attempts to look like Common Lisp but is weirdly and subtly missing all sorts of things. For instance:

(+ 1 2 3 4 5)

Is valid parenscript code, but

(apply #'+ '(1 2 3 4 5))

Will produce a run time error, because their is no + function in Parenscript. Similarly, list is just an alias for [] (which from a type perspective is ridiculous, the only thing arrays have in common with lists is that they are sequential) and so you can't apply list either. One of the goals of parenscript is to have no run time, but this is in direct contradiction to having parenscript look like Common Lisp.

There are plenty of Lisp dialects for the browser, as others have pointed out, (I am the author of one), but I wouldn't recommend most of them to anyone interested in just learning Lisp. They will introduce a layer of complexity that will make it more difficult than it needs to be.

As is usually the case when this comes up, I'd recommend programming in Racket which is a clean, modern Lisp dialect that will get you up to speed on Lispy ideas. Then maybe take a crack at Common Lisp. After all that, check out a Lisp dialect for the browser. It would be best to start with a Scheme dialect that compiles to Javascript because its likely that it will be complete and standard compliant, which means less confusing.